Integrand size = 31, antiderivative size = 654 \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {b g^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {b g^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}} \]
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Time = 0.77 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {4861, 4859, 4857, 3399, 4269, 3556, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {i g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {i g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {\sqrt {1-c^2 x^2} \tan \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))}{2 d \sqrt {d-c^2 d x^2} (c f+g)}-\frac {\sqrt {1-c^2 x^2} \cot \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right ) (a+b \arcsin (c x))}{2 d \sqrt {d-c^2 d x^2} (c f-g)}+\frac {b g^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {b g^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )\right )}{d \sqrt {d-c^2 d x^2} (c f-g)}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)+\frac {\pi }{4}\right )\right )}{d \sqrt {d-c^2 d x^2} (c f+g)} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3399
Rule 3404
Rule 3556
Rule 4269
Rule 4857
Rule 4859
Rule 4861
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{(f+g x) \left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = \frac {\sqrt {1-c^2 x^2} \int \left (-\frac {c (a+b \arcsin (c x))}{2 (c f+g) (-1+c x) \sqrt {1-c^2 x^2}}+\frac {c (a+b \arcsin (c x))}{2 (c f-g) (1+c x) \sqrt {1-c^2 x^2}}+\frac {g^2 (a+b \arcsin (c x))}{(-c f+g) (c f+g) (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{(1+c x) \sqrt {1-c^2 x^2}} \, dx}{2 d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{(-1+c x) \sqrt {1-c^2 x^2}} \, dx}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{d (-c f+g) (c f+g) \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{c+c \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{-c+c \sin (x)} \, dx,x,\arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{c f+g \sin (x)} \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \sqrt {d-c^2 d x^2}} \\ & = \frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x) \csc ^2\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{4 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x) \csc ^2\left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{4 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (2 g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \sqrt {d-c^2 d x^2}} \\ & = -\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \cot \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \cot \left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx,x,\arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\left (2 i g^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{2 c f-2 i e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 i g^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{2 c f-2 i e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \\ & = -\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (i b g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (i b g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arcsin (c x)\right )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \\ & = -\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {\left (b g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (b g^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d (-c f+g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \\ & = -\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \cot \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {i g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i g^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {b g^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {b g^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \tan \left (\frac {\pi }{4}+\frac {1}{2} \arcsin (c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 1.56 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.55 \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {1-c^2 x^2} \left (\frac {-\left ((a+b \arcsin (c x)) \cot \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+2 b \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )}{c f-g}+\frac {2 g^2 \left (i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-\log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )+b \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{(c f-g) (c f+g) \sqrt {c^2 f^2-g^2}}+\frac {2 b \log \left (\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+(a+b \arcsin (c x)) \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )}{c f+g}\right )}{2 d \sqrt {d-c^2 d x^2}} \]
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Time = 1.39 (sec) , antiderivative size = 1094, normalized size of antiderivative = 1.67
method | result | size |
default | \(\text {Expression too large to display}\) | \(1094\) |
parts | \(\text {Expression too large to display}\) | \(1094\) |
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\[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \]
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\[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (f + g x\right )}\, dx \]
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\[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \]
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Exception generated. \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {a+b \arcsin (c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{\left (f+g\,x\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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